If only Britain's pagan traditions weren't the preserve of hippies. Then we could all hold some sort of faintly debauched ceremony in Trafalgar Square and formally crown Lily Allen the Queen of Summer - a little like a May Queen, only hotter, less virginal, more mockney. It would be like England winning the Ashes, except for girls.

As the mercury has inexorably risen this summer, so has Lily. She began as a springtime music industry buzz that saw people unconnected with her record company raving about Allen's fabulousness to people like me. That doesn't normally happen.

It progressed at broadband speeds, as Allen's MySpace page became a venue of legend. Not only could you hear a few of her sharp, funny, ska-pop songs, you could read what Allen, a forthright and assiduous blogger, thought of certain indie rock stars, like her former schoolmate Luke Pritchard of the Kooks, and England's progress in the World Cup. 'Bob Geldof is a cunt,' it declares, randomly. More recently, she has posted pictures of her dog's new puppies, all being looked after by the parents of Allen's now-ex boyfriend in Ireland, for reasons undisclosed. (This is what the internet was invented for: unmediated glimpses into the lives of pop stars.)

The knowledge that Lily was the daughter of 'comedian' Keith Allen and a successful film producer arrived like a shower on this merry parade. It seemed Allen was just too good to be true after all, a Trustafarian brat hyped to the skies thanks to media nepotism. But as more songs emerged and 'Smile', her summery-sounding tune, stoked a fire under July, Lily sounded less and less like a jammy daddy's girl and more and more like the genuine article: a pop It Girl with the power to unite demographic tribes like no one since the Streets, to whom she has often been compared. Or the Arctic Monkeys, who kicked off this exceptional year for British music with a comparable reinvention of the musical wheel.

It's her second week at the top of the singles chart, and her debut album, Alright, Still, began flying out of shops earlier in the day. So it seems fitting that we should celebrate inside a hot cake. The snug, sweaty Bush Hall is a former dancehall bedecked with chandeliers and the rococo ceiling icing of architectural salvage shops. It normally hosts more sedate folk-type soirees, but the seats have been taken out to allow for mass skanking on the first of Allen's two sold-out nights. She is accompanied by a band - a drummer, keyboard player and bassist - and a three-piece brass section (trumpet, sax, trombone) whose glorious parping accentuates the Caribbean heat in Allen's songs. Madness spring to mind more than once. As does Gwen Stefani. You can imagine the multi-platinum American star being a teeny tiny bit jealous of a pop career so imbued by old ska records, Stefani's teenage passion. Unsurprisingly, they share a collaborator: Greg Kurstin, Stefani's live musical director, co-wrote three of Allen's tunes.

'I grew up two streets down from here,' Allen says, by way of introduction; everyone cheers. Now she towers over the area - literally. The Shepherd's Bush roundabout boasts a massive billboard advert for Allen's album: roadworks mean you can gaze at it for minutes on end.

In the flesh, Allen has yet to catch up with this stature. She may be wearing her signature ballgown, neon eyeshadow and dangly triangle earrings, like a Kelly Osbourne gone right, but Lily the teenager (she is just 21) is not far from the surface. Giggly and nervous, clutching a pint of cider and a cigarette, Allen grins and bobs her way through 'LDN', her ode to the capital's squalor. More than anything, it's a massive relief to hear she can hold a tune in public. Allen's voice isn't particularly muscular - the threat of being drowned out by the brass boys is ever-present tonight - but its lilting sweetness is real, not artificially enhanced by studio trickery. A wonky mike is dealt with efficiently, not spoiling 'Friday Night' too much.

Her real forte is chat. Allen's most riveting songs are delivered in a conversational patter, part West Indian tradition, part juicy gossip session. 'Knock 'Em Out' eavesdrops on an evening out spent jousting verbally with predatory men. It owes a great debt to the Streets. But Mike Skinner doesn't write from a female perspective. In fact, you can quite easily imagine a pestiferous Skinner being given a tongue-lashing by Allen. You are reminded, too, of Allen's fellow Londoner Lady Sovereign, with a sinking feeling. Sovereign is a more authentic, tomboyish MC whose sassy, chatted lyrics and sideways hair have suddenly been gazumped by this poppier, more bourgeois, better-connected arriviste in a party frock.

It would be a bigger shame if Allen wasn't such a hero. But she is. 'Not Big' is another matter-of-fact hammer blow to the privates of an unnamed ex. Pop knows no richer ground than that strewn with bits of broken heart, and Allen's revenge song is in the finest tradition of Eamon's 'Fuck It (I Don't Want You Back)' and choice man-dissing like TLC's 'No Scrubs'. Allen tops off her indictment - that the ex was rubbish in bed - with a wiggle of her little finger, Esperanto for 'tiddly winkle'. The backdrop lights up in sympathy.

Allen's between-song banter is almost as entertaining as the stuff she sets to music. She visibly relaxes as she introduces 'Littlest Things' as the song closest to her heart. This is the longest gig she has played yet, she confides. A whole hour! She takes a break halfway through to recover from the exertion. In all fairness, the heat is crippling. 'We love you, Lily!' someone heckles; 'I love myself!' she bats back, quick as you like.

If there is a downside to Allen's lippy charm, it's tonight's bass-heavy 'Nan, You're a Window Shopper', a parody of 50 Cent's 'Window Shopper' that was left off her album reportedly for reasons of sample clearance. But it could have been left off for its needlessly tasteless portrayal of an elderly woman, accused of being cheap, having bad taste in shopping carts, a leaking colostomy bag and cat hair on her clothes. You'd back Allen against almost anyone - her pothead younger brother Alfie (the subject of 'Alfie') is fair game - but a song about how old people smell is a waste of her talents. A question mark hangs over Allen's cover version of the Kaiser Chiefs' 'Oh My God', too. Allen's songs positively fizz with her personality. Singing someone else's stuff just doesn't suit her.

There is no encore. Not because of any punk rock attitude, or pop star stinginess, but because 'Oh My God' and 'Alfie' were meant to be the encore. She just forgets to walk off at the assigned time. Professionalism be damned: it's Lily Allen's summer, and she'll come and go as she wants to.